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Global controllability and stabilizability of Kawahara equation on a periodic domain
Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems
1. | GE Global Research, 1 Research Circle, Niskayuna, NY 12309, United States |
2. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074 |
3. | Department of Mathematics, Wayne State University, Detroit, Michigan 48202 |
References:
[1] |
J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations, Systems & Control Letters, 54 (2005), 1069-1081.
doi: 10.1016/j.sysconle.2005.03.003. |
[2] |
J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691.
doi: 10.1109/TAC.2008.919255. |
[3] |
L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise, SIAM J. Control Optim., 21 (1983), 451-461.
doi: 10.1137/0321027. |
[4] |
A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380.
doi: 10.1016/j.jmaa.2003.12.004. |
[5] |
F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth, Systems & Control Letters, 57 (2008), 262-270.
doi: 10.1016/j.sysconle.2007.09.002. |
[6] |
M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1952-1988.
doi: 10.1137/S0363012996299302. |
[7] |
R. Z. Khasminskii, Stochastic Stability of Differential Equations, $2^{nd}$ edition, Springer, Berlin, 2012.
doi: 10.1007/978-3-642-23280-0. |
[8] |
R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions, J. Differential Eqs., 161 (2000), 154-173.
doi: 10.1006/jdeq.1999.3647. |
[9] |
H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, $2^{nd}$ edition, Springer-Verlag, New York, NY, 2003.
doi: 10.1007/b97441. |
[10] |
B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339 (2008), 419-428.
doi: 10.1016/j.jmaa.2007.06.058. |
[11] |
R. Liptser and A. N. Shiryaev, Theory of Martingale, Kluwer Academic Publishers, Dordrecht, 1989.
doi: 10.1007/978-94-009-2438-3. |
[12] |
X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.
doi: 10.1016/S0304-4149(98)00070-2. |
[13] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.
doi: 10.1142/9781860948848_fmatter. |
[14] |
X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, UK, 2008.
doi: 10.1533/9780857099402. |
[15] |
A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Amer. Math. Soc., Providence, RI, 1989. |
[16] |
F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157.
doi: 10.1080/00207170902968108. |
[17] |
G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions, SIAM J. Math. Anal., 41 (2010), 2335-2352.
doi: 10.1137/080727191. |
[18] |
G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010.
doi: 10.1007/978-1-4419-1105-6. |
[19] |
G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382.
doi: 10.1137/110851171. |
[20] |
C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170.
doi: 10.1016/j.jmaa.2009.03.066. |
show all references
References:
[1] |
J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations, Systems & Control Letters, 54 (2005), 1069-1081.
doi: 10.1016/j.sysconle.2005.03.003. |
[2] |
J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691.
doi: 10.1109/TAC.2008.919255. |
[3] |
L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise, SIAM J. Control Optim., 21 (1983), 451-461.
doi: 10.1137/0321027. |
[4] |
A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380.
doi: 10.1016/j.jmaa.2003.12.004. |
[5] |
F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth, Systems & Control Letters, 57 (2008), 262-270.
doi: 10.1016/j.sysconle.2007.09.002. |
[6] |
M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), 1952-1988.
doi: 10.1137/S0363012996299302. |
[7] |
R. Z. Khasminskii, Stochastic Stability of Differential Equations, $2^{nd}$ edition, Springer, Berlin, 2012.
doi: 10.1007/978-3-642-23280-0. |
[8] |
R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions, J. Differential Eqs., 161 (2000), 154-173.
doi: 10.1006/jdeq.1999.3647. |
[9] |
H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, $2^{nd}$ edition, Springer-Verlag, New York, NY, 2003.
doi: 10.1007/b97441. |
[10] |
B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339 (2008), 419-428.
doi: 10.1016/j.jmaa.2007.06.058. |
[11] |
R. Liptser and A. N. Shiryaev, Theory of Martingale, Kluwer Academic Publishers, Dordrecht, 1989.
doi: 10.1007/978-94-009-2438-3. |
[12] |
X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.
doi: 10.1016/S0304-4149(98)00070-2. |
[13] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.
doi: 10.1142/9781860948848_fmatter. |
[14] |
X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, UK, 2008.
doi: 10.1533/9780857099402. |
[15] |
A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Amer. Math. Soc., Providence, RI, 1989. |
[16] |
F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157.
doi: 10.1080/00207170902968108. |
[17] |
G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions, SIAM J. Math. Anal., 41 (2010), 2335-2352.
doi: 10.1137/080727191. |
[18] |
G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010.
doi: 10.1007/978-1-4419-1105-6. |
[19] |
G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382.
doi: 10.1137/110851171. |
[20] |
C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170.
doi: 10.1016/j.jmaa.2009.03.066. |
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